src/HOL/TPTP/TPTP_Proof_Reconstruction.thy
author wenzelm
Fri Mar 06 15:58:56 2015 +0100 (2015-03-06)
changeset 59621 291934bac95e
parent 59582 0fbed69ff081
child 59639 f596ed647018
permissions -rw-r--r--
Thm.cterm_of and Thm.ctyp_of operate on local context;
     1 (*  Title:      HOL/TPTP/TPTP_Proof_Reconstruction.thy
     2     Author:     Nik Sultana, Cambridge University Computer Laboratory
     3 
     4 Proof reconstruction for Leo-II.
     5 
     6 USAGE:
     7 * Simple call the "reconstruct_leo2" function.
     8 * For more advanced use, you could use the component functions used in
     9   "reconstruct_leo2" -- see TPTP_Proof_Reconstruction_Test.thy for
    10   examples of this.
    11 
    12 This file contains definitions describing how to interpret LEO-II's
    13 calculus in Isabelle/HOL, as well as more general proof-handling
    14 functions. The definitions in this file serve to build an intermediate
    15 proof script which is then evaluated into a tactic -- both these steps
    16 are independent of LEO-II, and are defined in the TPTP_Reconstruct SML
    17 module.
    18 
    19 CONFIG:
    20 The following attributes are mainly useful for debugging:
    21   tptp_unexceptional_reconstruction |
    22   unexceptional_reconstruction      |-- when these are true, a low-level exception
    23                                         is allowed to float to the top (instead of
    24                                         triggering a higher-level exception, or
    25                                         simply indicating that the reconstruction failed).
    26   tptp_max_term_size                --- fail of a term exceeds this size. "0" is taken
    27                                         to mean infinity.
    28   tptp_informative_failure          |
    29   informative_failure               |-- produce more output during reconstruction.
    30   tptp_trace_reconstruction         |
    31 
    32 There are also two attributes, independent of the code here, that
    33 influence the success of reconstruction: blast_depth_limit and
    34 unify_search_bound. These are documented in their respective modules,
    35 but in summary, if unify_search_bound is increased then we can
    36 handle larger terms (at the cost of performance), since the unification
    37 engine takes longer to give up the search; blast_depth_limit is
    38 a limit on proof search performed by Blast. Blast is used for
    39 the limited proof search that needs to be done to interpret
    40 instances of LEO-II's inference rules.
    41 
    42 TODO:
    43   use RemoveRedundantQuantifications instead of the ad hoc use of
    44    remove_redundant_quantification_in_lit and remove_redundant_quantification
    45 *)
    46 
    47 theory TPTP_Proof_Reconstruction
    48 imports TPTP_Parser TPTP_Interpret
    49 (* keywords "import_leo2_proof" :: thy_decl *) (*FIXME currently unused*)
    50 begin
    51 
    52 
    53 section "Setup"
    54 
    55 ML {*
    56   val tptp_unexceptional_reconstruction = Attrib.setup_config_bool @{binding tptp_unexceptional_reconstruction} (K false)
    57   fun unexceptional_reconstruction ctxt = Config.get ctxt tptp_unexceptional_reconstruction
    58   val tptp_informative_failure = Attrib.setup_config_bool @{binding tptp_informative_failure} (K false)
    59   fun informative_failure ctxt = Config.get ctxt tptp_informative_failure
    60   val tptp_trace_reconstruction = Attrib.setup_config_bool @{binding tptp_trace_reconstruction} (K false)
    61   val tptp_max_term_size = Attrib.setup_config_int @{binding tptp_max_term_size} (K 0) (*0=infinity*)
    62 
    63   fun exceeds_tptp_max_term_size ctxt size =
    64     let
    65       val max = Config.get ctxt tptp_max_term_size
    66     in
    67       if max = 0 then false
    68       else size > max
    69     end
    70 *}
    71 
    72 (*FIXME move to TPTP_Proof_Reconstruction_Test_Units*)
    73 declare [[
    74   tptp_unexceptional_reconstruction = false, (*NOTE should be "false" while testing*)
    75   tptp_informative_failure = true
    76 ]]
    77 
    78 ML_file "TPTP_Parser/tptp_reconstruct_library.ML"
    79 ML "open TPTP_Reconstruct_Library"
    80 ML_file "TPTP_Parser/tptp_reconstruct.ML"
    81 
    82 (*FIXME fudge*)
    83 declare [[
    84   blast_depth_limit = 10,
    85   unify_search_bound = 5
    86 ]]
    87 
    88 
    89 section "Proof reconstruction"
    90 text {*There are two parts to proof reconstruction:
    91 \begin{itemize}
    92   \item interpreting the inferences
    93   \item building the skeleton, which indicates how to compose
    94     individual inferences into subproofs, and then compose the
    95     subproofs to give the proof).
    96 \end{itemize}
    97 
    98 One step detects unsound inferences, and the other step detects
    99 unsound composition of inferences.  The two parts can be weakly
   100 coupled. They rely on a "proof index" which maps nodes to the
   101 inference information. This information consists of the (usually
   102 prover-specific) name of the inference step, and the Isabelle
   103 formalisation of the inference as a term. The inference interpretation
   104 then maps these terms into meta-theorems, and the skeleton is used to
   105 compose the inference-level steps into a proof.
   106 
   107 Leo2 operates on conjunctions of clauses. Each Leo2 inference has the
   108 following form, where Cx are clauses:
   109 
   110            C1 && ... && Cn
   111           -----------------
   112           C'1 && ... && C'n
   113 
   114 Clauses consist of disjunctions of literals (shown as Px below), and might
   115 have a prefix of !-bound variables, as shown below.
   116 
   117   ! X... { P1 || ... || Pn}
   118 
   119 Literals are usually assigned a polarity, but this isn't always the
   120 case; you can come across inferences looking like this (where A is an
   121 object-level formula):
   122 
   123              F
   124           --------
   125           F = true
   126 
   127 The symbol "||" represents literal-level disjunction and "&&" is
   128 clause-level conjunction. Rules will typically lift formula-level
   129 conjunctions; for instance the following rule lifts object-level
   130 disjunction:
   131 
   132           {    (A | B) = true    || ... } && ...
   133           --------------------------------------
   134           { A = true || B = true || ... } && ...
   135 
   136 
   137 Using this setup, efficiency might be gained by only interpreting
   138 inferences once, merging identical inference steps, and merging
   139 identical subproofs into single inferences thus avoiding some effort.
   140 We can also attempt to minimising proof search when interpreting
   141 inferences.
   142 
   143 It is hoped that this setup can target other provers by modifying the
   144 clause representation to fit them, and adapting the inference
   145 interpretation to handle the rules used by the prover. It should also
   146 facilitate composing together proofs found by different provers.
   147 *}
   148 
   149 
   150 subsection "Instantiation"
   151 
   152 lemma polar_allE [rule_format]:
   153   "\<lbrakk>(\<forall>x. P x) = True; (P x) = True \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R"
   154   "\<lbrakk>(\<exists>x. P x) = False; (P x) = False \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R"
   155 by auto
   156 
   157 lemma polar_exE [rule_format]:
   158   "\<lbrakk>(\<exists>x. P x) = True; \<And>x. (P x) = True \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R"
   159   "\<lbrakk>(\<forall>x. P x) = False; \<And>x. (P x) = False \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R"
   160 by auto
   161 
   162 ML {*
   163 (*This carries out an allE-like rule but on (polarised) literals.
   164  Instead of yielding a free variable (which is a hell for the
   165  matcher) it seeks to use one of the subgoals' parameters.
   166  This ought to be sufficient for emulating extcnf_combined,
   167  but note that the complexity of the problem can be enormous.*)
   168 fun inst_parametermatch_tac ctxt thms i = fn st =>
   169   let
   170     val gls =
   171       Thm.prop_of st
   172       |> Logic.strip_horn
   173       |> fst
   174 
   175     val parameters =
   176       if null gls then []
   177       else
   178         rpair (i - 1) gls
   179         |> uncurry nth
   180         |> strip_top_all_vars []
   181         |> fst
   182         |> map fst (*just get the parameter names*)
   183   in
   184     if null parameters then no_tac st
   185     else
   186       let
   187         fun instantiate param =
   188            (map (eres_inst_tac ctxt [(("x", 0), param)]) thms
   189                    |> FIRST')
   190         val attempts = map instantiate parameters
   191       in
   192         (fold (curry (op APPEND')) attempts (K no_tac)) i st
   193       end
   194   end
   195 
   196 (*Attempts to use the polar_allE theorems on a specific subgoal.*)
   197 fun forall_pos_tac ctxt = inst_parametermatch_tac ctxt @{thms polar_allE}
   198 *}
   199 
   200 ML {*
   201 (*This is similar to inst_parametermatch_tac, but prefers to
   202   match variables having identical names. Logically, this is
   203   a hack. But it reduces the complexity of the problem.*)
   204 fun nominal_inst_parametermatch_tac ctxt thm i = fn st =>
   205   let
   206     val gls =
   207       Thm.prop_of st
   208       |> Logic.strip_horn
   209       |> fst
   210 
   211     val parameters =
   212       if null gls then []
   213       else
   214         rpair (i - 1) gls
   215         |> uncurry nth
   216         |> strip_top_all_vars []
   217         |> fst
   218         |> map fst (*just get the parameter names*)
   219   in
   220     if null parameters then no_tac st
   221     else
   222       let
   223         fun instantiates param =
   224            eres_inst_tac ctxt [(("x", 0), param)] thm
   225 
   226         val quantified_var = head_quantified_variable ctxt i st
   227       in
   228         if is_none quantified_var then no_tac st
   229         else
   230           if member (op =) parameters (the quantified_var |> fst) then
   231             instantiates (the quantified_var |> fst) i st
   232           else
   233             K no_tac i st
   234       end
   235   end
   236 *}
   237 
   238 
   239 subsection "Prefix massaging"
   240 
   241 ML {*
   242 exception NO_GOALS
   243 
   244 (*Get quantifier prefix of the hypothesis and conclusion, reorder
   245   the hypothesis' quantifiers to have the ones appearing in the
   246   conclusion first.*)
   247 fun canonicalise_qtfr_order ctxt i = fn st =>
   248   let
   249     val gls =
   250       Thm.prop_of st
   251       |> Logic.strip_horn
   252       |> fst
   253   in
   254     if null gls then raise NO_GOALS
   255     else
   256       let
   257         val (params, (hyp_clause, conc_clause)) =
   258           rpair (i - 1) gls
   259           |> uncurry nth
   260           |> strip_top_all_vars []
   261           |> apsnd Logic.dest_implies
   262 
   263         val (hyp_quants, hyp_body) =
   264           HOLogic.dest_Trueprop hyp_clause
   265           |> strip_top_All_vars
   266           |> apfst rev
   267 
   268         val conc_quants =
   269           HOLogic.dest_Trueprop conc_clause
   270           |> strip_top_All_vars
   271           |> fst
   272 
   273         val new_hyp =
   274           (* fold absfree new_hyp_prefix hyp_body *)
   275           (*HOLogic.list_all*)
   276           fold_rev (fn (v, ty) => fn t => HOLogic.mk_all (v, ty, t))
   277            (prefix_intersection_list
   278              hyp_quants conc_quants)
   279            hyp_body
   280           |> HOLogic.mk_Trueprop
   281 
   282          val thm = Goal.prove ctxt [] []
   283            (Logic.mk_implies (hyp_clause, new_hyp))
   284            (fn _ =>
   285               (REPEAT_DETERM (HEADGOAL (rtac @{thm allI})))
   286               THEN (REPEAT_DETERM
   287                     (HEADGOAL
   288                      (nominal_inst_parametermatch_tac ctxt @{thm allE})))
   289               THEN HEADGOAL atac)
   290       in
   291         dtac thm i st
   292       end
   293     end
   294 *}
   295 
   296 
   297 subsection "Some general rules and congruences"
   298 
   299 (*this isn't an actual rule used in Leo2, but it seems to be
   300   applied implicitly during some Leo2 inferences.*)
   301 lemma polarise: "P ==> P = True" by auto
   302 
   303 ML {*
   304 fun is_polarised t =
   305   (TPTP_Reconstruct.remove_polarity true t; true)
   306   handle TPTP_Reconstruct.UNPOLARISED _ => false
   307 
   308 val polarise_subgoal_hyps =
   309   COND' (SOME #> TERMPRED is_polarised (fn _ => true)) (K no_tac) (dtac @{thm polarise})
   310 *}
   311 
   312 lemma simp_meta [rule_format]:
   313   "(A --> B) == (~A | B)"
   314   "(A | B) | C == A | B | C"
   315   "(A & B) & C == A & B & C"
   316   "(~ (~ A)) == A"
   317   (* "(A & B) == (~ (~A | ~B))" *)
   318   "~ (A & B) == (~A | ~B)"
   319   "~(A | B) == (~A) & (~B)"
   320 by auto
   321 
   322 
   323 subsection "Emulation of Leo2's inference rules"
   324 
   325 (*this is not included in simp_meta since it would make a mess of the polarities*)
   326 lemma expand_iff [rule_format]:
   327  "((A :: bool) = B) \<equiv> (~ A | B) & (~ B | A)"
   328 by (rule eq_reflection, auto)
   329 
   330 lemma polarity_switch [rule_format]:
   331   "(\<not> P) = True \<Longrightarrow> P = False"
   332   "(\<not> P) = False \<Longrightarrow> P = True"
   333   "P = False \<Longrightarrow> (\<not> P) = True"
   334   "P = True \<Longrightarrow> (\<not> P) = False"
   335 by auto
   336 
   337 lemma solved_all_splits: "False = True \<Longrightarrow> False" by simp
   338 ML {*
   339 val solved_all_splits_tac =
   340   TRY (etac @{thm conjE} 1)
   341   THEN rtac @{thm solved_all_splits} 1
   342   THEN atac 1
   343 *}
   344 
   345 lemma lots_of_logic_expansions_meta [rule_format]:
   346   "(((A :: bool) = B) = True) == (((A \<longrightarrow> B) = True) & ((B \<longrightarrow> A) = True))"
   347   "((A :: bool) = B) = False == (((~A) | B) = False) | (((~B) | A) = False)"
   348 
   349   "((F = G) = True) == (! x. (F x = G x)) = True"
   350   "((F = G) = False) == (! x. (F x = G x)) = False"
   351 
   352   "(A | B) = True == (A = True) | (B = True)"
   353   "(A & B) = False == (A = False) | (B = False)"
   354   "(A | B) = False == (A = False) & (B = False)"
   355   "(A & B) = True == (A = True) & (B = True)"
   356   "(~ A) = True == A = False"
   357   "(~ A) = False == A = True"
   358   "~ (A = True) == A = False"
   359   "~ (A = False) == A = True"
   360 by (rule eq_reflection, auto)+
   361 
   362 (*this is used in extcnf_combined handler*)
   363 lemma eq_neg_bool: "((A :: bool) = B) = False ==> ((~ (A | B)) | ~ ((~ A) | (~ B))) = False"
   364 by auto
   365 
   366 lemma eq_pos_bool:
   367   "((A :: bool) = B) = True ==> ((~ (A | B)) | ~ (~ A | ~ B)) = True"
   368   "(A = B) = True \<Longrightarrow> A = True \<or> B = False"
   369   "(A = B) = True \<Longrightarrow> A = False \<or> B = True"
   370 by auto
   371 
   372 (*next formula is more versatile than
   373     "(F = G) = True \<Longrightarrow> \<forall>x. ((F x = G x) = True)"
   374   since it doesn't assume that clause is singleton. After splitqtfr,
   375   and after applying allI exhaustively to the conclusion, we can
   376   use the existing functions to find the "(F x = G x) = True"
   377   disjunct in the conclusion*)
   378 lemma eq_pos_func: "\<And> x. (F = G) = True \<Longrightarrow> (F x = G x) = True"
   379 by auto
   380 
   381 (*make sure the conclusion consists of just "False"*)
   382 lemma flip:
   383   "((A = True) ==> False) ==> A = False"
   384   "((A = False) ==> False) ==> A = True"
   385 by auto
   386 
   387 (*FIXME try to use Drule.equal_elim_rule1 directly for this*)
   388 lemma equal_elim_rule1: "(A \<equiv> B) \<Longrightarrow> A \<Longrightarrow> B" by auto
   389 lemmas leo2_rules =
   390  lots_of_logic_expansions_meta[THEN equal_elim_rule1]
   391 
   392 (*FIXME is there any overlap with lots_of_logic_expansions_meta or leo2_rules?*)
   393 lemma extuni_bool2 [rule_format]: "(A = B) = False \<Longrightarrow> (A = True) | (B = True)" by auto
   394 lemma extuni_bool1 [rule_format]: "(A = B) = False \<Longrightarrow> (A = False) | (B = False)" by auto
   395 lemma extuni_triv [rule_format]: "(A = A) = False \<Longrightarrow> R" by auto
   396 
   397 (*Order (of A, B, C, D) matters*)
   398 lemma dec_commut_eq [rule_format]:
   399   "((A = B) = (C = D)) = False \<Longrightarrow> (B = C) = False | (A = D) = False"
   400   "((A = B) = (C = D)) = False \<Longrightarrow> (B = D) = False | (A = C) = False"
   401 by auto
   402 lemma dec_commut_disj [rule_format]:
   403   "((A \<or> B) = (C \<or> D)) = False \<Longrightarrow> (B = C) = False \<or> (A = D) = False"
   404 by auto
   405 
   406 lemma extuni_func [rule_format]: "(F = G) = False \<Longrightarrow> (! X. (F X = G X)) = False" by auto
   407 
   408 
   409 subsection "Emulation: tactics"
   410 
   411 ML {*
   412 (*Instantiate a variable according to the info given in the
   413   proof annotation. Through this we avoid having to come up
   414   with instantiations during reconstruction.*)
   415 fun bind_tac ctxt prob_name ordered_binds =
   416   let
   417     val thy = Proof_Context.theory_of ctxt
   418     fun term_to_string t =
   419         Print_Mode.with_modes [""]
   420           (fn () => Output.output (Syntax.string_of_term ctxt t)) ()
   421     val ordered_instances =
   422       TPTP_Reconstruct.interpret_bindings prob_name thy ordered_binds []
   423       |> map (snd #> term_to_string)
   424       |> permute
   425 
   426     (*instantiate a list of variables, order matters*)
   427     fun instantiate_vars ctxt vars : tactic =
   428       map (fn var =>
   429             Rule_Insts.eres_inst_tac ctxt
   430              [(("x", 0), var)] @{thm allE} 1)
   431           vars
   432       |> EVERY
   433 
   434     fun instantiate_tac vars =
   435       instantiate_vars ctxt vars
   436       THEN (HEADGOAL atac)
   437   in
   438     HEADGOAL (canonicalise_qtfr_order ctxt)
   439     THEN (REPEAT_DETERM (HEADGOAL (rtac @{thm allI})))
   440     THEN REPEAT_DETERM (HEADGOAL (nominal_inst_parametermatch_tac ctxt @{thm allE}))
   441     (*now only the variable to instantiate should be left*)
   442     THEN FIRST (map instantiate_tac ordered_instances)
   443   end
   444 *}
   445 
   446 ML {*
   447 (*Simplification tactics*)
   448 local
   449   fun rew_goal_tac thms ctxt i =
   450     rewrite_goal_tac ctxt thms i
   451     |> CHANGED
   452 in
   453   val expander_animal =
   454     rew_goal_tac (@{thms simp_meta} @ @{thms lots_of_logic_expansions_meta})
   455 
   456   val simper_animal =
   457     rew_goal_tac @{thms simp_meta}
   458 end
   459 *}
   460 
   461 lemma prop_normalise [rule_format]:
   462   "(A | B) | C == A | B | C"
   463   "(A & B) & C == A & B & C"
   464   "A | B == ~(~A & ~B)"
   465   "~~ A == A"
   466 by auto
   467 ML {*
   468 (*i.e., break_conclusion*)
   469 fun flip_conclusion_tac ctxt =
   470   let
   471     val default_tac =
   472       (TRY o CHANGED o (rewrite_goal_tac ctxt @{thms prop_normalise}))
   473       THEN' rtac @{thm notI}
   474       THEN' (REPEAT_DETERM o etac @{thm conjE})
   475       THEN' (TRY o (expander_animal ctxt))
   476   in
   477     default_tac ORELSE' resolve_tac ctxt @{thms flip}
   478   end
   479 *}
   480 
   481 
   482 subsection "Skolemisation"
   483 
   484 lemma skolemise [rule_format]:
   485   "\<forall> P. (~ (! x. P x)) \<longrightarrow> ~ (P (SOME x. ~ P x))"
   486 proof -
   487   have "\<And> P. (~ (! x. P x)) \<Longrightarrow> ~ (P (SOME x. ~ P x))"
   488   proof -
   489     fix P
   490     assume ption: "~ (! x. P x)"
   491     hence a: "? x. ~ P x" by force
   492 
   493     have hilbert : "\<And> P. (? x. P x) \<Longrightarrow> (P (SOME x. P x))"
   494     proof -
   495       fix P
   496       assume "(? x. P x)"
   497       thus "(P (SOME x. P x))"
   498         apply auto
   499         apply (rule someI)
   500         apply auto
   501         done
   502     qed
   503 
   504     from a show "~ P (SOME x. ~ P x)"
   505     proof -
   506       assume "? x. ~ P x"
   507       hence "\<not> P (SOME x. \<not> P x)" by (rule hilbert)
   508       thus ?thesis .
   509     qed
   510   qed
   511   thus ?thesis by blast
   512 qed
   513 
   514 lemma polar_skolemise [rule_format]:
   515   "\<forall> P. (! x. P x) = False \<longrightarrow> (P (SOME x. ~ P x)) = False"
   516 proof -
   517   have "\<And> P. (! x. P x) = False \<Longrightarrow> (P (SOME x. ~ P x)) = False"
   518   proof -
   519     fix P
   520     assume ption: "(! x. P x) = False"
   521     hence "\<not> (\<forall> x. P x)" by force
   522     hence "\<not> All P" by force
   523     hence "\<not> (P (SOME x. \<not> P x))" by (rule skolemise)
   524     thus "(P (SOME x. \<not> P x)) = False" by force
   525   qed
   526   thus ?thesis by blast
   527 qed
   528 
   529 lemma leo2_skolemise [rule_format]:
   530   "\<forall> P sk. (! x. P x) = False \<longrightarrow> (sk = (SOME x. ~ P x)) \<longrightarrow> (P sk) = False"
   531 by (clarify, rule polar_skolemise)
   532 
   533 lemma lift_forall [rule_format]:
   534   "!! x. (! x. A x) = True ==> (A x) = True"
   535   "!! x. (? x. A x) = False ==> (A x) = False"
   536 by auto
   537 lemma lift_exists [rule_format]:
   538   "\<lbrakk>(All P) = False; sk = (SOME x. \<not> P x)\<rbrakk> \<Longrightarrow> P sk = False"
   539   "\<lbrakk>(Ex P) = True; sk = (SOME x. P x)\<rbrakk> \<Longrightarrow> P sk = True"
   540 apply (drule polar_skolemise, simp)
   541 apply (simp, drule someI_ex, simp)
   542 done
   543 
   544 ML {*
   545 (*FIXME LHS should be constant. Currently allow variables for testing. Probably should still allow Vars (but not Frees) since they'll act as intermediate values*)
   546 fun conc_is_skolem_def t =
   547   case t of
   548       Const (@{const_name HOL.eq}, _) $ t' $ (Const (@{const_name Hilbert_Choice.Eps}, _) $ _) =>
   549       let
   550         val (h, args) =
   551           strip_comb t'
   552           |> apfst (strip_abs #> snd #> strip_comb #> fst)
   553         val get_const_name = dest_Const #> fst
   554         val h_property =
   555           is_Free h orelse
   556           is_Var h orelse
   557           (is_Const h
   558            andalso (get_const_name h <> get_const_name @{term HOL.Ex})
   559            andalso (get_const_name h <> get_const_name @{term HOL.All})
   560            andalso (h <> @{term Hilbert_Choice.Eps})
   561            andalso (h <> @{term HOL.conj})
   562            andalso (h <> @{term HOL.disj})
   563            andalso (h <> @{term HOL.eq})
   564            andalso (h <> @{term HOL.implies})
   565            andalso (h <> @{term HOL.The})
   566            andalso (h <> @{term HOL.Ex1})
   567            andalso (h <> @{term HOL.Not})
   568            andalso (h <> @{term HOL.iff})
   569            andalso (h <> @{term HOL.not_equal}))
   570         val args_property =
   571           fold (fn t => fn b =>
   572            b andalso is_Free t) args true
   573       in
   574         h_property andalso args_property
   575       end
   576     | _ => false
   577 *}
   578 
   579 ML {*
   580 (*Hack used to detect if a Skolem definition, with an LHS Var, has had the LHS instantiated into an unacceptable term.*)
   581 fun conc_is_bad_skolem_def t =
   582   case t of
   583       Const (@{const_name HOL.eq}, _) $ t' $ (Const (@{const_name Hilbert_Choice.Eps}, _) $ _) =>
   584       let
   585         val (h, args) = strip_comb t'
   586         val get_const_name = dest_Const #> fst
   587         val const_h_test =
   588           if is_Const h then
   589             (get_const_name h = get_const_name @{term HOL.Ex})
   590              orelse (get_const_name h = get_const_name @{term HOL.All})
   591              orelse (h = @{term Hilbert_Choice.Eps})
   592              orelse (h = @{term HOL.conj})
   593              orelse (h = @{term HOL.disj})
   594              orelse (h = @{term HOL.eq})
   595              orelse (h = @{term HOL.implies})
   596              orelse (h = @{term HOL.The})
   597              orelse (h = @{term HOL.Ex1})
   598              orelse (h = @{term HOL.Not})
   599              orelse (h = @{term HOL.iff})
   600              orelse (h = @{term HOL.not_equal})
   601           else true
   602         val h_property =
   603           not (is_Free h) andalso
   604           not (is_Var h) andalso
   605           const_h_test
   606         val args_property =
   607           fold (fn t => fn b =>
   608            b andalso is_Free t) args true
   609       in
   610         h_property andalso args_property
   611       end
   612     | _ => false
   613 *}
   614 
   615 ML {*
   616 fun get_skolem_conc t =
   617   let
   618     val t' =
   619       strip_top_all_vars [] t
   620       |> snd
   621       |> try_dest_Trueprop
   622   in
   623     case t' of
   624         Const (@{const_name HOL.eq}, _) $ t' $ (Const (@{const_name Hilbert_Choice.Eps}, _) $ _) => SOME t'
   625       | _ => NONE
   626   end
   627 
   628 fun get_skolem_conc_const t =
   629   lift_option
   630    (fn t' =>
   631      head_of t'
   632      |> strip_abs_body
   633      |> head_of
   634      |> dest_Const)
   635    (get_skolem_conc t)
   636 *}
   637 
   638 (*
   639 Technique for handling quantifiers:
   640   Principles:
   641   * allE should always match with a !!
   642   * exE should match with a constant,
   643      or bind a fresh !! -- currently not doing the latter since it never seems to arised in normal Leo2 proofs.
   644 *)
   645 
   646 ML {*
   647 fun forall_neg_tac candidate_consts ctxt i = fn st =>
   648   let
   649     val thy = Proof_Context.theory_of ctxt
   650 
   651     val gls =
   652       Thm.prop_of st
   653       |> Logic.strip_horn
   654       |> fst
   655 
   656     val parameters =
   657       if null gls then ""
   658       else
   659         rpair (i - 1) gls
   660         |> uncurry nth
   661         |> strip_top_all_vars []
   662         |> fst
   663         |> map fst (*just get the parameter names*)
   664         |> (fn l =>
   665               if null l then ""
   666               else
   667                 space_implode " " l
   668                 |> pair " "
   669                 |> op ^)
   670 
   671   in
   672     if null gls orelse null candidate_consts then no_tac st
   673     else
   674       let
   675         fun instantiate const_name =
   676           dres_inst_tac ctxt [(("sk", 0), const_name ^ parameters)] @{thm leo2_skolemise}
   677         val attempts = map instantiate candidate_consts
   678       in
   679         (fold (curry (op APPEND')) attempts (K no_tac)) i st
   680       end
   681   end
   682 *}
   683 
   684 ML {*
   685 exception SKOLEM_DEF of term (*The tactic wasn't pointed at a skolem definition*)
   686 exception NO_SKOLEM_DEF of (*skolem const name*)string * Binding.binding * term (*The tactic could not find a skolem definition in the theory*)
   687 fun absorb_skolem_def ctxt prob_name_opt i = fn st =>
   688   let
   689     val thy = Proof_Context.theory_of ctxt
   690 
   691     val gls =
   692       Thm.prop_of st
   693       |> Logic.strip_horn
   694       |> fst
   695 
   696     val conclusion =
   697       if null gls then
   698         (*this should never be thrown*)
   699         raise NO_GOALS
   700       else
   701         rpair (i - 1) gls
   702         |> uncurry nth
   703         |> strip_top_all_vars []
   704         |> snd
   705         |> Logic.strip_horn
   706         |> snd
   707 
   708     fun skolem_const_info_of t =
   709       case t of
   710           Const (@{const_name HOL.Trueprop}, _) $ (Const (@{const_name HOL.eq}, _) $ t' $ (Const (@{const_name Hilbert_Choice.Eps}, _) $ _)) =>
   711           head_of t'
   712           |> strip_abs_body (*since in general might have a skolem term, so we want to rip out the prefixing lambdas to get to the constant (which should be at head position)*)
   713           |> head_of
   714           |> dest_Const
   715         | _ => raise SKOLEM_DEF t
   716 
   717     val const_name =
   718       skolem_const_info_of conclusion
   719       |> fst
   720 
   721     val def_name = const_name ^ "_def"
   722 
   723     val bnd_def = (*FIXME consts*)
   724       const_name
   725       |> Long_Name.implode o tl o Long_Name.explode (*FIXME hack to drop theory-name prefix*)
   726       |> Binding.qualified_name
   727       |> Binding.suffix_name "_def"
   728 
   729     val bnd_name =
   730       case prob_name_opt of
   731           NONE => bnd_def
   732         | SOME prob_name =>
   733 (*            Binding.qualify false
   734              (TPTP_Problem_Name.mangle_problem_name prob_name)
   735 *)
   736              bnd_def
   737 
   738     val thm =
   739       if Name_Space.defined_entry (Theory.axiom_space thy) def_name then
   740         Thm.axiom thy def_name
   741       else
   742         if is_none prob_name_opt then
   743           (*This mode is for testing, so we can be a bit
   744             looser with theories*)
   745           Thm.add_axiom_global (bnd_name, conclusion) thy
   746           |> fst |> snd
   747         else
   748           raise (NO_SKOLEM_DEF (def_name, bnd_name, conclusion))
   749   in
   750     rtac (Drule.export_without_context thm) i st
   751   end
   752   handle SKOLEM_DEF _ => no_tac st
   753 *}
   754 
   755 ML {*
   756 (*
   757 In current system, there should only be 2 subgoals: the one where
   758 the skolem definition is being built (with a Var in the LHS), and the other subgoal using Var.
   759 *)
   760 (*arity must be greater than 0. if arity=0 then
   761   there's no need to use this expensive matching.*)
   762 fun find_skolem_term ctxt consts_candidate arity = fn st =>
   763   let
   764     val _ = @{assert} (arity > 0)
   765 
   766     val gls =
   767       Thm.prop_of st
   768       |> Logic.strip_horn
   769       |> fst
   770 
   771     (*extract the conclusion of each subgoal*)
   772     val conclusions =
   773       if null gls then
   774         raise NO_GOALS
   775       else
   776         map (strip_top_all_vars [] #> snd #> Logic.strip_horn #> snd) gls
   777         (*Remove skolem-definition conclusion, to avoid wasting time analysing it*)
   778         |> filter (try_dest_Trueprop #> conc_is_skolem_def #> not)
   779         (*There should only be a single goal*) (*FIXME this might not always be the case, in practice*)
   780         (* |> tap (fn x => @{assert} (is_some (try the_single x))) *)
   781 
   782     (*look for subterms headed by a skolem constant, and whose
   783       arguments are all parameter Vars*)
   784     fun get_skolem_terms args (acc : term list) t =
   785       case t of
   786           (c as Const _) $ (v as Free _) =>
   787             if c = consts_candidate andalso
   788              arity = length args + 1 then
   789               (list_comb (c, v :: args)) :: acc
   790             else acc
   791         | t1 $ (v as Free _) =>
   792             get_skolem_terms (v :: args) acc t1 @
   793              get_skolem_terms [] acc t1
   794         | t1 $ t2 =>
   795             get_skolem_terms [] acc t1 @
   796              get_skolem_terms [] acc t2
   797         | Abs (_, _, t') => get_skolem_terms [] acc t'
   798         | _ => acc
   799   in
   800     map (strip_top_All_vars #> snd) conclusions
   801     |> maps (get_skolem_terms [] [])
   802     |> distinct (op =)
   803   end
   804 *}
   805 
   806 ML {*
   807 fun instantiate_skols ctxt consts_candidates i = fn st =>
   808   let
   809     val thy = Proof_Context.theory_of ctxt
   810 
   811     val gls =
   812       Thm.prop_of st
   813       |> Logic.strip_horn
   814       |> fst
   815 
   816     val (params, conclusion) =
   817       if null gls then
   818         raise NO_GOALS
   819       else
   820         rpair (i - 1) gls
   821         |> uncurry nth
   822         |> strip_top_all_vars []
   823         |> apsnd (Logic.strip_horn #> snd)
   824 
   825     fun skolem_const_info_of t =
   826       case t of
   827           Const (@{const_name HOL.Trueprop}, _) $ (Const (@{const_name HOL.eq}, _) $ lhs $ (Const (@{const_name Hilbert_Choice.Eps}, _) $ rhs)) =>
   828           let
   829             (*the parameters we will concern ourselves with*)
   830             val params' =
   831               Term.add_frees lhs []
   832               |> distinct (op =)
   833             (*check to make sure that params' <= params*)
   834             val _ = @{assert} (forall (member (op =) params) params')
   835             val skolem_const_ty =
   836               let
   837                 val (skolem_const_prety, no_params) =
   838                   Term.strip_comb lhs
   839                   |> apfst (dest_Var #> snd) (*head of lhs consists of a logical variable. we just want its type.*)
   840                   |> apsnd length
   841 
   842                 val _ = @{assert} (length params = no_params)
   843 
   844                 (*get value type of a function type after n arguments have been supplied*)
   845                 fun get_val_ty n ty =
   846                   if n = 0 then ty
   847                   else get_val_ty (n - 1) (dest_funT ty |> snd)
   848               in
   849                 get_val_ty no_params skolem_const_prety
   850               end
   851 
   852           in
   853             (skolem_const_ty, params')
   854           end
   855         | _ => raise (SKOLEM_DEF t)
   856 
   857 (*
   858 find skolem const candidates which, after applying distinct members of params' we end up with, give us something of type skolem_const_ty.
   859 
   860 given a candidate's type, skolem_const_ty, and params', we get some pemutations of params' (i.e. the order in which they can be given to the candidate in order to get skolem_const_ty). If the list of permutations is empty, then we cannot use that candidate.
   861 *)
   862 (*
   863 only returns a single matching -- since terms are linear, and variable arguments are Vars, order shouldn't matter, so we can ignore permutations.
   864 doesn't work with polymorphism (for which we'd need to use type unification) -- this is OK since no terms should be polymorphic, since Leo2 proofs aren't.
   865 *)
   866     fun use_candidate target_ty params acc cur_ty =
   867       if null params then
   868         if cur_ty = target_ty then
   869           SOME (rev acc)
   870         else NONE
   871       else
   872         let
   873           val (arg_ty, val_ty) = Term.dest_funT cur_ty
   874           (*now find a param of type arg_ty*)
   875           val (candidate_param, params') =
   876             find_and_remove (snd #> pair arg_ty #> op =) params
   877         in
   878           use_candidate target_ty params' (candidate_param :: acc) val_ty
   879         end
   880         handle TYPE ("dest_funT", _, _) => NONE
   881              | DEST_LIST => NONE
   882 
   883     val (skolem_const_ty, params') = skolem_const_info_of conclusion
   884 
   885 (*
   886 For each candidate, build a term and pass it to Thm.instantiate, whic in turn is chained with PRIMITIVE to give us this_tactic.
   887 
   888 Big picture:
   889   we run the following:
   890     drule leo2_skolemise THEN' this_tactic
   891 
   892 NOTE: remember to APPEND' instead of ORELSE' the two tactics relating to skolemisation
   893 *)
   894 
   895     val filtered_candidates =
   896       map (dest_Const
   897            #> snd
   898            #> use_candidate skolem_const_ty params' [])
   899        consts_candidates (* prefiltered_candidates *)
   900       |> pair consts_candidates (* prefiltered_candidates *)
   901       |> ListPair.zip
   902       |> filter (snd #> is_none #> not)
   903       |> map (apsnd the)
   904 
   905     val skolem_terms =
   906       let
   907         fun make_result_t (t, args) =
   908           (* list_comb (t, map Free args) *)
   909           if length args > 0 then
   910             hd (find_skolem_term ctxt t (length args) st)
   911           else t
   912       in
   913         map make_result_t filtered_candidates
   914       end
   915 
   916     (*prefix a skolem term with bindings for the parameters*)
   917     (* val contextualise = fold absdummy (map snd params) *)
   918     val contextualise = fold absfree params
   919 
   920     val skolem_cts = map (contextualise #> Thm.global_cterm_of thy) skolem_terms
   921 
   922 
   923 (*now the instantiation code*)
   924 
   925     (*there should only be one Var -- that is from the previous application of drule leo2_skolemise. We look for it at the head position in some equation at a conclusion of a subgoal.*)
   926     val var_opt =
   927       let
   928         val pre_var =
   929           gls
   930           |> map
   931               (strip_top_all_vars [] #> snd #>
   932                Logic.strip_horn #> snd #>
   933                get_skolem_conc)
   934           |> switch (fold (fn x => fn l => if is_some x then the x :: l else l)) []
   935           |> maps (switch Term.add_vars [])
   936 
   937         fun make_var pre_var =
   938           the_single pre_var
   939           |> Var
   940           |> Thm.global_cterm_of thy
   941           |> SOME
   942       in
   943         if null pre_var then NONE
   944         else make_var pre_var
   945      end
   946 
   947     fun instantiate_tac from to =
   948       Thm.instantiate ([], [(from, to)])
   949       |> PRIMITIVE
   950 
   951     val tectic =
   952       if is_none var_opt then no_tac
   953       else
   954         fold (curry (op APPEND)) (map (instantiate_tac (the var_opt)) skolem_cts) no_tac
   955 
   956   in
   957     tectic st
   958   end
   959 *}
   960 
   961 ML {*
   962 fun new_skolem_tac ctxt consts_candidates =
   963   let
   964     fun tec thm =
   965       dtac thm
   966       THEN' instantiate_skols ctxt consts_candidates
   967   in
   968     if null consts_candidates then K no_tac
   969     else FIRST' (map tec @{thms lift_exists})
   970   end
   971 *}
   972 
   973 (*
   974 need a tactic to expand "? x . P" to "~ ! x. ~ P"
   975 *)
   976 ML {*
   977 fun ex_expander_tac ctxt i =
   978    let
   979      val simpset =
   980        empty_simpset ctxt (*NOTE for some reason, Bind exception gets raised if ctxt's simpset isn't emptied*)
   981        |> Simplifier.add_simp @{lemma "Ex P == (~ (! x. ~ P x))" by auto}
   982    in
   983      CHANGED (asm_full_simp_tac simpset i)
   984    end
   985 *}
   986 
   987 
   988 subsubsection "extuni_dec"
   989 
   990 ML {*
   991 (*n-ary decomposition. Code is based on the n-ary arg_cong generator*)
   992 fun extuni_dec_n ctxt arity =
   993   let
   994     val _ = @{assert} (arity > 0)
   995     val is =
   996       upto (1, arity)
   997       |> map Int.toString
   998     val arg_tys = map (fn i => TFree ("arg" ^ i ^ "_ty", @{sort type})) is
   999     val res_ty = TFree ("res" ^ "_ty", @{sort type})
  1000     val f_ty = arg_tys ---> res_ty
  1001     val f = Free ("f", f_ty)
  1002     val xs = map (fn i =>
  1003       Free ("x" ^ i, TFree ("arg" ^ i ^ "_ty", @{sort type}))) is
  1004     (*FIXME DRY principle*)
  1005     val ys = map (fn i =>
  1006       Free ("y" ^ i, TFree ("arg" ^ i ^ "_ty", @{sort type}))) is
  1007 
  1008     val hyp_lhs = list_comb (f, xs)
  1009     val hyp_rhs = list_comb (f, ys)
  1010     val hyp_eq =
  1011       HOLogic.eq_const res_ty $ hyp_lhs $ hyp_rhs
  1012     val hyp =
  1013       HOLogic.eq_const HOLogic.boolT $ hyp_eq $ @{term False}
  1014       |> HOLogic.mk_Trueprop
  1015     fun conc_eq i =
  1016       let
  1017         val ty = TFree ("arg" ^ i ^ "_ty", @{sort type})
  1018         val x = Free ("x" ^ i, ty)
  1019         val y = Free ("y" ^ i, ty)
  1020         val eq = HOLogic.eq_const ty $ x $ y
  1021       in
  1022         HOLogic.eq_const HOLogic.boolT $ eq $ @{term False}
  1023       end
  1024 
  1025     val conc_disjs = map conc_eq is
  1026 
  1027     val conc =
  1028       if length conc_disjs = 1 then
  1029         the_single conc_disjs
  1030       else
  1031         fold
  1032          (fn t => fn t_conc => HOLogic.mk_disj (t_conc, t))
  1033          (tl conc_disjs) (hd conc_disjs)
  1034 
  1035     val t =
  1036       Logic.mk_implies (hyp, HOLogic.mk_Trueprop conc)
  1037   in
  1038     Goal.prove ctxt [] [] t (fn _ => auto_tac ctxt)
  1039     |> Drule.export_without_context
  1040   end
  1041 *}
  1042 
  1043 ML {*
  1044 (*Determine the arity of a function which the "dec"
  1045   unification rule is about to be applied.
  1046   NOTE:
  1047     * Assumes that there is a single hypothesis
  1048 *)
  1049 fun find_dec_arity i = fn st =>
  1050   let
  1051     val gls =
  1052       Thm.prop_of st
  1053       |> Logic.strip_horn
  1054       |> fst
  1055   in
  1056     if null gls then raise NO_GOALS
  1057     else
  1058       let
  1059         val (params, (literal, conc_clause)) =
  1060           rpair (i - 1) gls
  1061           |> uncurry nth
  1062           |> strip_top_all_vars []
  1063           |> apsnd Logic.strip_horn
  1064           |> apsnd (apfst the_single)
  1065 
  1066         val get_ty =
  1067           HOLogic.dest_Trueprop
  1068           #> strip_top_All_vars
  1069           #> snd
  1070           #> HOLogic.dest_eq (*polarity's "="*)
  1071           #> fst
  1072           #> HOLogic.dest_eq (*the unification constraint's "="*)
  1073           #> fst
  1074           #> head_of
  1075           #> dest_Const
  1076           #> snd
  1077 
  1078        fun arity_of ty =
  1079          let
  1080            val (_, res_ty) = dest_funT ty
  1081 
  1082          in
  1083            1 + arity_of res_ty
  1084          end
  1085          handle (TYPE ("dest_funT", _, _)) => 0
  1086 
  1087       in
  1088         arity_of (get_ty literal)
  1089       end
  1090   end
  1091 
  1092 (*given an inference, it returns the parameters (i.e., we've already matched the leading & shared quantification in the hypothesis & conclusion clauses), and the "raw" inference*)
  1093 fun breakdown_inference i = fn st =>
  1094   let
  1095     val gls =
  1096       Thm.prop_of st
  1097       |> Logic.strip_horn
  1098       |> fst
  1099   in
  1100     if null gls then raise NO_GOALS
  1101     else
  1102       rpair (i - 1) gls
  1103       |> uncurry nth
  1104       |> strip_top_all_vars []
  1105   end
  1106 
  1107 (*build a custom elimination rule for extuni_dec, and instantiate it to match a specific subgoal*)
  1108 fun extuni_dec_elim_rule ctxt arity i = fn st =>
  1109   let
  1110     val rule = extuni_dec_n ctxt arity
  1111 
  1112     val rule_hyp =
  1113       Thm.prop_of rule
  1114       |> Logic.dest_implies
  1115       |> fst (*assuming that rule has single hypothesis*)
  1116 
  1117     (*having run break_hypothesis earlier, we know that the hypothesis
  1118       now consists of a single literal. We can (and should)
  1119       disregard the conclusion, since it hasn't been "broken",
  1120       and it might include some unwanted literals -- the latter
  1121       could cause "diff" to fail (since they won't agree with the
  1122       rule we have generated.*)
  1123 
  1124     val inference_hyp =
  1125       snd (breakdown_inference i st)
  1126       |> Logic.dest_implies
  1127       |> fst (*assuming that inference has single hypothesis,
  1128                as explained above.*)
  1129   in
  1130     TPTP_Reconstruct_Library.diff_and_instantiate ctxt rule rule_hyp inference_hyp
  1131   end
  1132 
  1133 fun extuni_dec_tac ctxt i = fn st =>
  1134   let
  1135     val arity = find_dec_arity i st
  1136 
  1137     fun elim_tac i st =
  1138       let
  1139         val rule =
  1140           extuni_dec_elim_rule ctxt arity i st
  1141           (*in case we itroduced free variables during
  1142             instantiation, we generalise the rule to make
  1143             those free variables into logical variables.*)
  1144           |> Thm.forall_intr_frees
  1145           |> Drule.export_without_context
  1146       in dtac rule i st end
  1147       handle NO_GOALS => no_tac st
  1148 
  1149     fun closure tac =
  1150      (*batter fails if there's no toplevel disjunction in the
  1151        hypothesis, so we also try atac*)
  1152       SOLVE o (tac THEN' (batter_tac ctxt ORELSE' assume_tac ctxt))
  1153     val search_tac =
  1154       ASAP
  1155         (rtac @{thm disjI1} APPEND' rtac @{thm disjI2})
  1156         (FIRST' (map closure
  1157                   [dresolve_tac ctxt @{thms dec_commut_eq},
  1158                    dtac @{thm dec_commut_disj},
  1159                    elim_tac]))
  1160   in
  1161     (CHANGED o search_tac) i st
  1162   end
  1163 *}
  1164 
  1165 
  1166 subsubsection "standard_cnf"
  1167 (*Given a standard_cnf inference, normalise it
  1168      e.g. ((A & B) & C \<longrightarrow> D & E \<longrightarrow> F \<longrightarrow> G) = False
  1169      is changed to
  1170           (A & B & C & D & E & F \<longrightarrow> G) = False
  1171  then custom-build a metatheorem which validates this:
  1172           (A & B & C & D & E & F \<longrightarrow> G) = False
  1173        -------------------------------------------
  1174           (A = True) & (B = True) & (C = True) &
  1175           (D = True) & (E = True) & (F = True) & (G = False)
  1176  and apply this metatheorem.
  1177 
  1178 There aren't any "positive" standard_cnfs in Leo2's calculus:
  1179   e.g.,  "(A \<longrightarrow> B) = True \<Longrightarrow> A = False | (A = True & B = True)"
  1180 since "standard_cnf" seems to be applied at the preprocessing
  1181 stage, together with splitting.
  1182 *)
  1183 
  1184 ML {*
  1185 (*Conjunctive counterparts to Term.disjuncts_aux and Term.disjuncts*)
  1186 fun conjuncts_aux (Const (@{const_name HOL.conj}, _) $ t $ t') conjs =
  1187      conjuncts_aux t (conjuncts_aux t' conjs)
  1188   | conjuncts_aux t conjs = t :: conjs
  1189 
  1190 fun conjuncts t = conjuncts_aux t []
  1191 
  1192 (*HOL equivalent of Logic.strip_horn*)
  1193 local
  1194   fun imp_strip_horn' acc (Const (@{const_name HOL.implies}, _) $ A $ B) =
  1195         imp_strip_horn' (A :: acc) B
  1196     | imp_strip_horn' acc t = (acc, t)
  1197 in
  1198   fun imp_strip_horn t =
  1199     imp_strip_horn' [] t
  1200     |> apfst rev
  1201 end
  1202 *}
  1203 
  1204 ML {*
  1205 (*Returns whether the antecedents are separated by conjunctions
  1206   or implications; the number of antecedents; and the polarity
  1207   of the original clause -- I think this will always be "false".*)
  1208 fun standard_cnf_type ctxt i : thm -> (TPTP_Reconstruct.formula_kind * int * bool) option = fn st =>
  1209   let
  1210     val gls =
  1211       Thm.prop_of st
  1212       |> Logic.strip_horn
  1213       |> fst
  1214 
  1215     val hypos =
  1216       if null gls then raise NO_GOALS
  1217       else
  1218         rpair (i - 1) gls
  1219         |> uncurry nth
  1220         |> TPTP_Reconstruct.strip_top_all_vars []
  1221         |> snd
  1222         |> Logic.strip_horn
  1223         |> fst
  1224 
  1225     (*hypothesis clause should be singleton*)
  1226     val _ = @{assert} (length hypos = 1)
  1227 
  1228     val (t, pol) = the_single hypos
  1229       |> try_dest_Trueprop
  1230       |> TPTP_Reconstruct.strip_top_All_vars
  1231       |> snd
  1232       |> TPTP_Reconstruct.remove_polarity true
  1233 
  1234     (*literal is negative*)
  1235     val _ = @{assert} (not pol)
  1236 
  1237     val (antes, conc) = imp_strip_horn t
  1238 
  1239     val (ante_type, antes') =
  1240       if length antes = 1 then
  1241         let
  1242           val conjunctive_antes =
  1243             the_single antes
  1244             |> conjuncts
  1245         in
  1246           if length conjunctive_antes > 1 then
  1247             (TPTP_Reconstruct.Conjunctive NONE,
  1248              conjunctive_antes)
  1249           else
  1250             (TPTP_Reconstruct.Implicational NONE,
  1251              antes)
  1252         end
  1253       else
  1254         (TPTP_Reconstruct.Implicational NONE,
  1255          antes)
  1256   in
  1257     if null antes then NONE
  1258     else SOME (ante_type, length antes', pol)
  1259   end
  1260 *}
  1261 
  1262 ML {*
  1263 (*Given a certain standard_cnf type, build a metatheorem that would
  1264   validate it*)
  1265 fun mk_standard_cnf ctxt kind arity =
  1266   let
  1267     val _ = @{assert} (arity > 0)
  1268     val vars =
  1269       upto (1, arity + 1)
  1270       |> map (fn i => Free ("x" ^ Int.toString i, HOLogic.boolT))
  1271 
  1272     val consequent = hd vars
  1273     val antecedents = tl vars
  1274 
  1275     val conc =
  1276       fold
  1277        (curry HOLogic.mk_conj)
  1278        (map (fn var => HOLogic.mk_eq (var, @{term True})) antecedents)
  1279        (HOLogic.mk_eq (consequent, @{term False}))
  1280 
  1281     val pre_hyp =
  1282       case kind of
  1283           TPTP_Reconstruct.Conjunctive NONE =>
  1284             curry HOLogic.mk_imp
  1285              (if length antecedents = 1 then the_single antecedents
  1286               else
  1287                 fold (curry HOLogic.mk_conj) (tl antecedents) (hd antecedents))
  1288              (hd vars)
  1289         | TPTP_Reconstruct.Implicational NONE =>
  1290             fold (curry HOLogic.mk_imp) antecedents consequent
  1291 
  1292     val hyp = HOLogic.mk_eq (pre_hyp, @{term False})
  1293 
  1294     val t =
  1295       Logic.mk_implies (HOLogic.mk_Trueprop  hyp, HOLogic.mk_Trueprop conc)
  1296   in
  1297     Goal.prove ctxt [] [] t (fn _ => HEADGOAL (blast_tac ctxt))
  1298     |> Drule.export_without_context
  1299   end
  1300 *}
  1301 
  1302 ML {*
  1303 (*Applies a d-tactic, then breaks it up conjunctively.
  1304   This can be used to transform subgoals as follows:
  1305      (A \<longrightarrow> B) = False  \<Longrightarrow> R
  1306               |
  1307               v
  1308   \<lbrakk>A = True; B = False\<rbrakk> \<Longrightarrow> R
  1309 *)
  1310 fun weak_conj_tac drule =
  1311   dtac drule THEN' (REPEAT_DETERM o etac @{thm conjE})
  1312 *}
  1313 
  1314 ML {*
  1315 val uncurry_lit_neg_tac =
  1316   dtac @{lemma "(A \<longrightarrow> B \<longrightarrow> C) = False \<Longrightarrow> (A & B \<longrightarrow> C) = False" by auto}
  1317   #> REPEAT_DETERM
  1318 *}
  1319 
  1320 ML {*
  1321 fun standard_cnf_tac ctxt i = fn st =>
  1322   let
  1323     fun core_tactic i = fn st =>
  1324       case standard_cnf_type ctxt i st of
  1325           NONE => no_tac st
  1326         | SOME (kind, arity, _) =>
  1327             let
  1328               val rule = mk_standard_cnf ctxt kind arity;
  1329             in
  1330               (weak_conj_tac rule THEN' atac) i st
  1331             end
  1332   in
  1333     (uncurry_lit_neg_tac
  1334      THEN' TPTP_Reconstruct_Library.reassociate_conjs_tac ctxt
  1335      THEN' core_tactic) i st
  1336   end
  1337 *}
  1338 
  1339 
  1340 subsubsection "Emulator prep"
  1341 
  1342 ML {*
  1343 datatype cleanup_feature =
  1344     RemoveHypothesesFromSkolemDefs
  1345   | RemoveDuplicates
  1346 
  1347 datatype loop_feature =
  1348     Close_Branch
  1349   | ConjI
  1350   | King_Cong
  1351   | Break_Hypotheses
  1352   | Donkey_Cong (*simper_animal + ex_expander_tac*)
  1353   | RemoveRedundantQuantifications
  1354   | Assumption
  1355 
  1356   (*Closely based on Leo2 calculus*)
  1357   | Existential_Free
  1358   | Existential_Var
  1359   | Universal
  1360   | Not_pos
  1361   | Not_neg
  1362   | Or_pos
  1363   | Or_neg
  1364   | Equal_pos
  1365   | Equal_neg
  1366   | Extuni_Bool2
  1367   | Extuni_Bool1
  1368   | Extuni_Dec
  1369   | Extuni_Bind
  1370   | Extuni_Triv
  1371   | Extuni_FlexRigid
  1372   | Extuni_Func
  1373   | Polarity_switch
  1374   | Forall_special_pos
  1375 
  1376 datatype feature =
  1377     ConstsDiff
  1378   | StripQuantifiers
  1379   | Flip_Conclusion
  1380   | Loop of loop_feature list
  1381   | LoopOnce of loop_feature list
  1382   | InnerLoopOnce of loop_feature list
  1383   | CleanUp of cleanup_feature list
  1384   | AbsorbSkolemDefs
  1385 *}
  1386 
  1387 ML {*
  1388 fun can_feature x l =
  1389   let
  1390     fun sublist_of_clean_up el =
  1391       case el of
  1392           CleanUp l'' => SOME l''
  1393         | _ => NONE
  1394     fun sublist_of_loop el =
  1395       case el of
  1396           Loop l'' => SOME l''
  1397         | _ => NONE
  1398     fun sublist_of_loop_once el =
  1399       case el of
  1400           LoopOnce l'' => SOME l''
  1401         | _ => NONE
  1402     fun sublist_of_inner_loop_once el =
  1403       case el of
  1404           InnerLoopOnce l'' => SOME l''
  1405         | _ => NONE
  1406 
  1407     fun check_sublist sought_sublist opt_list =
  1408       if forall is_none opt_list then false
  1409       else
  1410         fold_options opt_list
  1411         |> flat
  1412         |> pair sought_sublist
  1413         |> subset (op =)
  1414   in
  1415     case x of
  1416         CleanUp l' =>
  1417           map sublist_of_clean_up l
  1418           |> check_sublist l'
  1419       | Loop l' =>
  1420           map sublist_of_loop l
  1421           |> check_sublist l'
  1422       | LoopOnce l' =>
  1423           map sublist_of_loop_once l
  1424           |> check_sublist l'
  1425       | InnerLoopOnce l' =>
  1426           map sublist_of_inner_loop_once l
  1427           |> check_sublist l'
  1428       | _ => exists (curry (op =) x) l
  1429   end;
  1430 
  1431 fun loop_can_feature loop_feats l =
  1432   can_feature (Loop loop_feats) l orelse
  1433   can_feature (LoopOnce loop_feats) l orelse
  1434   can_feature (InnerLoopOnce loop_feats) l;
  1435 
  1436 @{assert} (can_feature ConstsDiff [StripQuantifiers, ConstsDiff]);
  1437 
  1438 @{assert}
  1439   (can_feature (CleanUp [RemoveHypothesesFromSkolemDefs])
  1440     [CleanUp [RemoveHypothesesFromSkolemDefs, RemoveDuplicates]]);
  1441 
  1442 @{assert}
  1443   (can_feature (Loop []) [Loop [Existential_Var]]);
  1444 
  1445 @{assert}
  1446   (not (can_feature (Loop []) [InnerLoopOnce [Existential_Var]]));
  1447 *}
  1448 
  1449 ML {*
  1450 exception NO_LOOP_FEATS
  1451 fun get_loop_feats (feats : feature list) =
  1452   let
  1453     val loop_find =
  1454       fold (fn x => fn loop_feats_acc =>
  1455         if is_some loop_feats_acc then loop_feats_acc
  1456         else
  1457           case x of
  1458               Loop loop_feats => SOME loop_feats
  1459             | LoopOnce loop_feats => SOME loop_feats
  1460             | InnerLoopOnce loop_feats => SOME loop_feats
  1461             | _ => NONE)
  1462        feats
  1463        NONE
  1464   in
  1465     if is_some loop_find then the loop_find
  1466     else raise NO_LOOP_FEATS
  1467   end;
  1468 
  1469 @{assert}
  1470   (get_loop_feats [Loop [King_Cong, Break_Hypotheses, Existential_Free, Existential_Var, Universal]] =
  1471    [King_Cong, Break_Hypotheses, Existential_Free, Existential_Var, Universal])
  1472 *}
  1473 
  1474 (*use as elim rule to remove premises*)
  1475 lemma insa_prems: "\<lbrakk>Q; P\<rbrakk> \<Longrightarrow> P" by auto
  1476 ML {*
  1477 fun cleanup_skolem_defs feats =
  1478   let
  1479     (*remove hypotheses from skolem defs,
  1480      after testing that they look like skolem defs*)
  1481     val dehypothesise_skolem_defs =
  1482       COND' (SOME #> TERMPRED (fn _ => true) conc_is_skolem_def)
  1483         (REPEAT_DETERM o etac @{thm insa_prems})
  1484         (K no_tac)
  1485   in
  1486     if can_feature (CleanUp [RemoveHypothesesFromSkolemDefs]) feats then
  1487       ALLGOALS (TRY o dehypothesise_skolem_defs)
  1488     else all_tac
  1489   end
  1490 *}
  1491 
  1492 ML {*
  1493 fun remove_duplicates_tac feats =
  1494   (if can_feature (CleanUp [RemoveDuplicates]) feats then
  1495      ALLGOALS distinct_subgoal_tac
  1496    else all_tac)
  1497 *}
  1498 
  1499 ML {*
  1500 (*given a goal state, indicates the skolem constants committed-to in it (i.e. appearing in LHS of a skolem definition)*)
  1501 val which_skolem_concs_used = fn st =>
  1502   let
  1503     val feats = [CleanUp [RemoveHypothesesFromSkolemDefs, RemoveDuplicates]]
  1504     val scrubup_tac =
  1505       cleanup_skolem_defs feats
  1506       THEN remove_duplicates_tac feats
  1507   in
  1508     scrubup_tac st
  1509     |> break_seq
  1510     |> tap (fn (_, rest) => @{assert} (null (Seq.list_of rest)))
  1511     |> fst
  1512     |> TERMFUN (snd (*discard hypotheses*)
  1513                  #> get_skolem_conc_const) NONE
  1514     |> switch (fold (fn x => fn l => if is_some x then the x :: l else l)) []
  1515     |> map Const
  1516   end
  1517 *}
  1518 
  1519 ML {*
  1520 fun exists_tac ctxt feats consts_diff =
  1521   let
  1522     val ex_var =
  1523       if loop_can_feature [Existential_Var] feats andalso consts_diff <> [] then
  1524         new_skolem_tac ctxt consts_diff
  1525         (*We're making sure that each skolem constant is used once in instantiations.*)
  1526       else K no_tac
  1527 
  1528     val ex_free =
  1529       if loop_can_feature [Existential_Free] feats andalso consts_diff = [] then
  1530         eresolve_tac ctxt @{thms polar_exE}
  1531       else K no_tac
  1532   in
  1533     ex_var APPEND' ex_free
  1534   end
  1535 
  1536 fun forall_tac ctxt feats =
  1537   if loop_can_feature [Universal] feats then
  1538     forall_pos_tac ctxt
  1539   else K no_tac
  1540 *}
  1541 
  1542 
  1543 subsubsection "Finite types"
  1544 (*lift quantification from a singleton literal to a singleton clause*)
  1545 lemma forall_pos_lift:
  1546 "\<lbrakk>(! X. P X) = True; ! X. (P X = True) \<Longrightarrow> R\<rbrakk> \<Longrightarrow> R" by auto
  1547 
  1548 (*predicate over the type of the leading quantified variable*)
  1549 
  1550 ML {*
  1551 val extcnf_forall_special_pos_tac =
  1552   let
  1553     val bool =
  1554       ["True", "False"]
  1555 
  1556     val bool_to_bool =
  1557       ["% _ . True", "% _ . False", "% x . x", "Not"]
  1558 
  1559     val tecs =
  1560       map (fn t_s =>
  1561        eres_inst_tac @{context} [(("x", 0), t_s)] @{thm allE}
  1562        THEN' atac)
  1563   in
  1564     (TRY o etac @{thm forall_pos_lift})
  1565     THEN' (atac
  1566            ORELSE' FIRST'
  1567             (*FIXME could check the type of the leading quantified variable, instead of trying everything*)
  1568             (tecs (bool @ bool_to_bool)))
  1569   end
  1570 *}
  1571 
  1572 
  1573 subsubsection "Emulator"
  1574 
  1575 lemma efq: "[|A = True; A = False|] ==> R" by auto
  1576 ML {*
  1577 val efq_tac =
  1578   (etac @{thm efq} THEN' atac)
  1579   ORELSE' atac
  1580 *}
  1581 
  1582 ML {*
  1583 (*This is applied to all subgoals, repeatedly*)
  1584 fun extcnf_combined_main ctxt feats consts_diff =
  1585   let
  1586     (*This is applied to subgoals which don't have a conclusion
  1587       consisting of a Skolem definition*)
  1588     fun extcnf_combined_tac' ctxt i = fn st =>
  1589       let
  1590         val skolem_consts_used_so_far = which_skolem_concs_used st
  1591         val consts_diff' = subtract (op =) skolem_consts_used_so_far consts_diff
  1592 
  1593         fun feat_to_tac feat =
  1594           case feat of
  1595               Close_Branch => trace_tac' ctxt "mark: closer" efq_tac
  1596             | ConjI => trace_tac' ctxt "mark: conjI" (rtac @{thm conjI})
  1597             | King_Cong => trace_tac' ctxt "mark: expander_animal" (expander_animal ctxt)
  1598             | Break_Hypotheses => trace_tac' ctxt "mark: break_hypotheses" (break_hypotheses_tac ctxt)
  1599             | RemoveRedundantQuantifications => K all_tac
  1600 (*
  1601 FIXME Building this into the loop instead.. maybe not the ideal choice
  1602             | RemoveRedundantQuantifications =>
  1603                 trace_tac' ctxt "mark: strip_unused_variable_hyp"
  1604                  (REPEAT_DETERM o remove_redundant_quantification_in_lit)
  1605 *)
  1606 
  1607             | Assumption => atac
  1608 (*FIXME both Existential_Free and Existential_Var run same code*)
  1609             | Existential_Free => trace_tac' ctxt "mark: forall_neg" (exists_tac ctxt feats consts_diff')
  1610             | Existential_Var => trace_tac' ctxt "mark: forall_neg" (exists_tac ctxt feats consts_diff')
  1611             | Universal => trace_tac' ctxt "mark: forall_pos" (forall_tac ctxt feats)
  1612             | Not_pos => trace_tac' ctxt "mark: not_pos" (dtac @{thm leo2_rules(9)})
  1613             | Not_neg => trace_tac' ctxt "mark: not_neg" (dtac @{thm leo2_rules(10)})
  1614             | Or_pos => trace_tac' ctxt "mark: or_pos" (dtac @{thm leo2_rules(5)}) (*could add (6) for negated conjunction*)
  1615             | Or_neg => trace_tac' ctxt "mark: or_neg" (dtac @{thm leo2_rules(7)})
  1616             | Equal_pos => trace_tac' ctxt "mark: equal_pos" (dresolve_tac ctxt (@{thms eq_pos_bool} @ [@{thm leo2_rules(3)}, @{thm eq_pos_func}]))
  1617             | Equal_neg => trace_tac' ctxt "mark: equal_neg" (dresolve_tac ctxt [@{thm eq_neg_bool}, @{thm leo2_rules(4)}])
  1618             | Donkey_Cong => trace_tac' ctxt "mark: donkey_cong" (simper_animal ctxt THEN' ex_expander_tac ctxt)
  1619 
  1620             | Extuni_Bool2 => trace_tac' ctxt "mark: extuni_bool2" (dtac @{thm extuni_bool2})
  1621             | Extuni_Bool1 => trace_tac' ctxt "mark: extuni_bool1" (dtac @{thm extuni_bool1})
  1622             | Extuni_Bind => trace_tac' ctxt "mark: extuni_triv" (etac @{thm extuni_triv})
  1623             | Extuni_Triv => trace_tac' ctxt "mark: extuni_triv" (etac @{thm extuni_triv})
  1624             | Extuni_Dec => trace_tac' ctxt "mark: extuni_dec_tac" (extuni_dec_tac ctxt)
  1625             | Extuni_FlexRigid => trace_tac' ctxt "mark: extuni_flex_rigid" (atac ORELSE' asm_full_simp_tac ctxt)
  1626             | Extuni_Func => trace_tac' ctxt "mark: extuni_func" (dtac @{thm extuni_func})
  1627             | Polarity_switch => trace_tac' ctxt "mark: polarity_switch" (eresolve_tac ctxt @{thms polarity_switch})
  1628             | Forall_special_pos => trace_tac' ctxt "mark: dorall_special_pos" extcnf_forall_special_pos_tac
  1629 
  1630         val core_tac =
  1631           get_loop_feats feats
  1632           |> map feat_to_tac
  1633           |> FIRST'
  1634       in
  1635         core_tac i st
  1636       end
  1637 
  1638     (*This is applied to all subgoals, repeatedly*)
  1639     fun extcnf_combined_tac ctxt i =
  1640       COND (TERMPRED (fn _ => true) conc_is_skolem_def (SOME i))
  1641         no_tac
  1642         (extcnf_combined_tac' ctxt i)
  1643 
  1644     val core_tac = CHANGED (ALLGOALS (IF_UNSOLVED o TRY o extcnf_combined_tac ctxt))
  1645 
  1646     val full_tac = REPEAT core_tac
  1647 
  1648   in
  1649     CHANGED
  1650       (if can_feature (InnerLoopOnce []) feats then
  1651          core_tac
  1652        else full_tac)
  1653   end
  1654 
  1655 val interpreted_consts =
  1656   [@{const_name HOL.All}, @{const_name HOL.Ex},
  1657    @{const_name Hilbert_Choice.Eps},
  1658    @{const_name HOL.conj},
  1659    @{const_name HOL.disj},
  1660    @{const_name HOL.eq},
  1661    @{const_name HOL.implies},
  1662    @{const_name HOL.The},
  1663    @{const_name HOL.Ex1},
  1664    @{const_name HOL.Not},
  1665    (* @{const_name HOL.iff}, *) (*FIXME do these exist?*)
  1666    (* @{const_name HOL.not_equal}, *)
  1667    @{const_name HOL.False},
  1668    @{const_name HOL.True},
  1669    @{const_name Pure.imp}]
  1670 
  1671 fun strip_qtfrs_tac ctxt =
  1672   REPEAT_DETERM (HEADGOAL (rtac @{thm allI}))
  1673   THEN REPEAT_DETERM (HEADGOAL (etac @{thm exE}))
  1674   THEN HEADGOAL (canonicalise_qtfr_order ctxt)
  1675   THEN
  1676     ((REPEAT (HEADGOAL (nominal_inst_parametermatch_tac ctxt @{thm allE})))
  1677      APPEND (REPEAT (HEADGOAL (inst_parametermatch_tac ctxt [@{thm allE}]))))
  1678   (*FIXME need to handle "@{thm exI}"?*)
  1679 
  1680 (*difference in constants between the hypothesis clause and the conclusion clause*)
  1681 fun clause_consts_diff thm =
  1682   let
  1683     val t =
  1684       Thm.prop_of thm
  1685       |> Logic.dest_implies
  1686       |> fst
  1687 
  1688       (*This bit should not be needed, since Leo2 inferences don't have parameters*)
  1689       |> TPTP_Reconstruct.strip_top_all_vars []
  1690       |> snd
  1691 
  1692     val do_diff =
  1693       Logic.dest_implies
  1694       #> uncurry TPTP_Reconstruct.new_consts_between
  1695       #> filter
  1696            (fn Const (n, _) =>
  1697              not (member (op =) interpreted_consts n))
  1698   in
  1699     if head_of t = Logic.implies then do_diff t
  1700     else []
  1701   end
  1702 *}
  1703 
  1704 ML {*
  1705 (*remove quantification in hypothesis clause (! X. t), if
  1706   X not free in t*)
  1707 fun remove_redundant_quantification ctxt i = fn st =>
  1708   let
  1709     val gls =
  1710       Thm.prop_of st
  1711       |> Logic.strip_horn
  1712       |> fst
  1713   in
  1714     if null gls then raise NO_GOALS
  1715     else
  1716       let
  1717         val (params, (hyp_clauses, conc_clause)) =
  1718           rpair (i - 1) gls
  1719           |> uncurry nth
  1720           |> TPTP_Reconstruct.strip_top_all_vars []
  1721           |> apsnd Logic.strip_horn
  1722       in
  1723         (*this is to fail gracefully in case this tactic is applied to a goal which doesn't have a single hypothesis*)
  1724         if length hyp_clauses > 1 then no_tac st
  1725         else
  1726           let
  1727             val hyp_clause = the_single hyp_clauses
  1728             val sep_prefix =
  1729               HOLogic.dest_Trueprop
  1730               #> TPTP_Reconstruct.strip_top_All_vars
  1731               #> apfst rev
  1732             val (hyp_prefix, hyp_body) = sep_prefix hyp_clause
  1733             val (conc_prefix, conc_body) = sep_prefix conc_clause
  1734           in
  1735             if null hyp_prefix orelse
  1736               member (op =) conc_prefix (hd hyp_prefix) orelse
  1737               member (op =)  (Term.add_frees hyp_body []) (hd hyp_prefix) then
  1738               no_tac st
  1739             else
  1740               eres_inst_tac ctxt [(("x", 0), "(@X. False)")] @{thm allE} i st
  1741           end
  1742      end
  1743   end
  1744 *}
  1745 
  1746 ML {*
  1747 fun remove_redundant_quantification_ignore_skolems ctxt i =
  1748   COND (TERMPRED (fn _ => true) conc_is_skolem_def (SOME i))
  1749     no_tac
  1750     (remove_redundant_quantification ctxt i)
  1751 *}
  1752 
  1753 lemma drop_redundant_literal_qtfr:
  1754   "(! X. P) = True \<Longrightarrow> P = True"
  1755   "(? X. P) = True \<Longrightarrow> P = True"
  1756   "(! X. P) = False \<Longrightarrow> P = False"
  1757   "(? X. P) = False \<Longrightarrow> P = False"
  1758 by auto
  1759 
  1760 ML {*
  1761 (*remove quantification in the literal "(! X. t) = True/False"
  1762   in the singleton hypothesis clause, if X not free in t*)
  1763 fun remove_redundant_quantification_in_lit ctxt i = fn st =>
  1764   let
  1765     val gls =
  1766       Thm.prop_of st
  1767       |> Logic.strip_horn
  1768       |> fst
  1769   in
  1770     if null gls then raise NO_GOALS
  1771     else
  1772       let
  1773         val (params, (hyp_clauses, conc_clause)) =
  1774           rpair (i - 1) gls
  1775           |> uncurry nth
  1776           |> TPTP_Reconstruct.strip_top_all_vars []
  1777           |> apsnd Logic.strip_horn
  1778       in
  1779         (*this is to fail gracefully in case this tactic is applied to a goal which doesn't have a single hypothesis*)
  1780         if length hyp_clauses > 1 then no_tac st
  1781         else
  1782           let
  1783             fun literal_content (Const (@{const_name HOL.eq}, _) $ lhs $ (rhs as @{term True})) = SOME (lhs, rhs)
  1784               | literal_content (Const (@{const_name HOL.eq}, _) $ lhs $ (rhs as @{term False})) = SOME (lhs, rhs)
  1785               | literal_content t = NONE
  1786 
  1787             val hyp_clause =
  1788               the_single hyp_clauses
  1789               |> HOLogic.dest_Trueprop
  1790               |> literal_content
  1791 
  1792           in
  1793             if is_none hyp_clause then
  1794               no_tac st
  1795             else
  1796               let
  1797                 val (hyp_lit_prefix, hyp_lit_body) =
  1798                   the hyp_clause
  1799                   |> (fn (t, polarity) =>
  1800                        TPTP_Reconstruct.strip_top_All_vars t
  1801                        |> apfst rev)
  1802               in
  1803                 if null hyp_lit_prefix orelse
  1804                   member (op =)  (Term.add_frees hyp_lit_body []) (hd hyp_lit_prefix) then
  1805                   no_tac st
  1806                 else
  1807                   dresolve_tac ctxt @{thms drop_redundant_literal_qtfr} i st
  1808               end
  1809           end
  1810      end
  1811   end
  1812 *}
  1813 
  1814 ML {*
  1815 fun remove_redundant_quantification_in_lit_ignore_skolems ctxt i =
  1816   COND (TERMPRED (fn _ => true) conc_is_skolem_def (SOME i))
  1817     no_tac
  1818     (remove_redundant_quantification_in_lit ctxt i)
  1819 *}
  1820 
  1821 ML {*
  1822 fun extcnf_combined_tac ctxt prob_name_opt feats skolem_consts = fn st =>
  1823   let
  1824     val thy = Proof_Context.theory_of ctxt
  1825 
  1826     (*Initially, st consists of a single goal, showing the
  1827       hypothesis clause implying the conclusion clause.
  1828       There are no parameters.*)
  1829     val consts_diff =
  1830       union (op =) skolem_consts
  1831        (if can_feature ConstsDiff feats then
  1832           clause_consts_diff st
  1833         else [])
  1834 
  1835     val main_tac =
  1836       if can_feature (LoopOnce []) feats orelse can_feature (InnerLoopOnce []) feats then
  1837         extcnf_combined_main ctxt feats consts_diff
  1838       else if can_feature (Loop []) feats then
  1839         BEST_FIRST (TERMPRED (fn _ => true) conc_is_skolem_def NONE, size_of_thm)
  1840 (*FIXME maybe need to weaken predicate to include "solved form"?*)
  1841          (extcnf_combined_main ctxt feats consts_diff)
  1842       else all_tac (*to allow us to use the cleaning features*)
  1843 
  1844     (*Remove hypotheses from Skolem definitions,
  1845       then remove duplicate subgoals,
  1846       then we should be left with skolem definitions:
  1847         absorb them as axioms into the theory.*)
  1848     val cleanup =
  1849       cleanup_skolem_defs feats
  1850       THEN remove_duplicates_tac feats
  1851       THEN (if can_feature AbsorbSkolemDefs feats then
  1852               ALLGOALS (absorb_skolem_def ctxt prob_name_opt)
  1853             else all_tac)
  1854 
  1855     val have_loop_feats =
  1856       (get_loop_feats feats; true)
  1857       handle NO_LOOP_FEATS => false
  1858 
  1859     val tec =
  1860       (if can_feature StripQuantifiers feats then
  1861          (REPEAT (CHANGED (strip_qtfrs_tac ctxt)))
  1862        else all_tac)
  1863       THEN (if can_feature Flip_Conclusion feats then
  1864              HEADGOAL (flip_conclusion_tac ctxt)
  1865            else all_tac)
  1866 
  1867       (*after stripping the quantifiers any remaining quantifiers
  1868         can be simply eliminated -- they're redundant*)
  1869       (*FIXME instead of just using allE, instantiate to a silly
  1870          term, to remove opportunities for unification.*)
  1871       THEN (REPEAT_DETERM (etac @{thm allE} 1))
  1872 
  1873       THEN (REPEAT_DETERM (rtac @{thm allI} 1))
  1874 
  1875       THEN (if have_loop_feats then
  1876               REPEAT (CHANGED
  1877               ((ALLGOALS (TRY o clause_breaker_tac ctxt)) (*brush away literals which don't change*)
  1878                THEN
  1879                 (*FIXME move this to a different level?*)
  1880                 (if loop_can_feature [Polarity_switch] feats then
  1881                    all_tac
  1882                  else
  1883                    (TRY (IF_UNSOLVED (HEADGOAL (remove_redundant_quantification_ignore_skolems ctxt))))
  1884                    THEN (TRY (IF_UNSOLVED (HEADGOAL (remove_redundant_quantification_in_lit_ignore_skolems ctxt)))))
  1885                THEN (TRY main_tac)))
  1886             else
  1887               all_tac)
  1888       THEN IF_UNSOLVED cleanup
  1889 
  1890   in
  1891     DEPTH_SOLVE (CHANGED tec) st
  1892   end
  1893 *}
  1894 
  1895 
  1896 subsubsection "unfold_def"
  1897 
  1898 (*this is used when handling unfold_tac, because the skeleton includes the definitions conjoined with the goal. it turns out that, for my tactic, the definitions are harmful. instead of modifying the skeleton (which may be nontrivial) i'm just dropping the information using this lemma. obviously, and from the name, order matters here.*)
  1899 lemma drop_first_hypothesis [rule_format]: "\<lbrakk>A; B\<rbrakk> \<Longrightarrow> B" by auto
  1900 
  1901 (*Unfold_def works by reducing the goal to a meta equation,
  1902   then working on it until it can be discharged by atac,
  1903   or reflexive, or else turned back into an object equation
  1904   and broken down further.*)
  1905 lemma un_meta_polarise: "(X \<equiv> True) \<Longrightarrow> X" by auto
  1906 lemma meta_polarise: "X \<Longrightarrow> X \<equiv> True" by auto
  1907 
  1908 ML {*
  1909 fun unfold_def_tac ctxt depends_on_defs = fn st =>
  1910   let
  1911     (*This is used when we end up with something like
  1912         (A & B) \<equiv> True \<Longrightarrow> (B & A) \<equiv> True.
  1913       It breaks down this subgoal until it can be trivially
  1914       discharged.
  1915      *)
  1916     val kill_meta_eqs_tac =
  1917       dtac @{thm un_meta_polarise}
  1918       THEN' rtac @{thm meta_polarise}
  1919       THEN' (REPEAT_DETERM o (etac @{thm conjE}))
  1920       THEN' (REPEAT_DETERM o (rtac @{thm conjI} ORELSE' atac))
  1921 
  1922     val continue_reducing_tac =
  1923       rtac @{thm meta_eq_to_obj_eq} 1
  1924       THEN (REPEAT_DETERM (ex_expander_tac ctxt 1))
  1925       THEN TRY (polarise_subgoal_hyps 1) (*no need to REPEAT_DETERM here, since there should only be one hypothesis*)
  1926       THEN TRY (dtac @{thm eq_reflection} 1)
  1927       THEN (TRY ((CHANGED o rewrite_goal_tac ctxt
  1928               (@{thm expand_iff} :: @{thms simp_meta})) 1))
  1929       THEN HEADGOAL (rtac @{thm reflexive}
  1930                      ORELSE' atac
  1931                      ORELSE' kill_meta_eqs_tac)
  1932 
  1933     val tectic =
  1934       (rtac @{thm polarise} 1 THEN atac 1)
  1935       ORELSE
  1936         (REPEAT_DETERM (etac @{thm conjE} 1 THEN etac @{thm drop_first_hypothesis} 1)
  1937          THEN PRIMITIVE (Conv.fconv_rule Drule.eta_long_conversion)
  1938          THEN (REPEAT_DETERM (ex_expander_tac ctxt 1))
  1939          THEN (TRY ((CHANGED o rewrite_goal_tac ctxt @{thms simp_meta}) 1))
  1940          THEN PRIMITIVE (Conv.fconv_rule Drule.eta_long_conversion)
  1941          THEN
  1942            (HEADGOAL atac
  1943            ORELSE
  1944             (unfold_tac ctxt depends_on_defs
  1945              THEN IF_UNSOLVED continue_reducing_tac)))
  1946   in
  1947     tectic st
  1948   end
  1949 *}
  1950 
  1951 
  1952 subsection "Handling split 'preprocessing'"
  1953 
  1954 lemma split_tranfs:
  1955   "! x. P x & Q x \<equiv> (! x. P x) & (! x. Q x)"
  1956   "~ (~ A) \<equiv> A"
  1957   "? x. A \<equiv> A"
  1958   "(A & B) & C \<equiv> A & B & C"
  1959   "A = B \<equiv> (A --> B) & (B --> A)"
  1960 by (rule eq_reflection, auto)+
  1961 
  1962 (*Same idiom as ex_expander_tac*)
  1963 ML {*
  1964 fun split_simp_tac (ctxt : Proof.context) i =
  1965    let
  1966      val simpset =
  1967        fold Simplifier.add_simp @{thms split_tranfs} (empty_simpset ctxt)
  1968    in
  1969      CHANGED (asm_full_simp_tac simpset i)
  1970    end
  1971 *}
  1972 
  1973 
  1974 subsection "Alternative reconstruction tactics"
  1975 ML {*
  1976 (*An "auto"-based proof reconstruction, where we attempt to reconstruct each inference
  1977   using auto_tac. A realistic tactic would inspect the inference name and act
  1978   accordingly.*)
  1979 fun auto_based_reconstruction_tac ctxt prob_name n =
  1980   let
  1981     val thy = Proof_Context.theory_of ctxt
  1982     val pannot = TPTP_Reconstruct.get_pannot_of_prob thy prob_name
  1983   in
  1984     TPTP_Reconstruct.inference_at_node
  1985      thy
  1986      prob_name (#meta pannot) n
  1987       |> the
  1988       |> (fn {inference_fmla, ...} =>
  1989           Goal.prove ctxt [] [] inference_fmla
  1990            (fn pdata => auto_tac (#context pdata)))
  1991   end
  1992 *}
  1993 
  1994 (*An oracle-based reconstruction, which is only used to test the shunting part of the system*)
  1995 oracle oracle_iinterp = "fn t => t"
  1996 ML {*
  1997 fun oracle_based_reconstruction_tac ctxt prob_name n =
  1998   let
  1999     val thy = Proof_Context.theory_of ctxt
  2000     val pannot = TPTP_Reconstruct.get_pannot_of_prob thy prob_name
  2001   in
  2002     TPTP_Reconstruct.inference_at_node
  2003      thy
  2004      prob_name (#meta pannot) n
  2005       |> the
  2006       |> (fn {inference_fmla, ...} => Thm.global_cterm_of thy inference_fmla)
  2007       |> oracle_iinterp
  2008   end
  2009 *}
  2010 
  2011 
  2012 subsection "Leo2 reconstruction tactic"
  2013 
  2014 ML {*
  2015 exception UNSUPPORTED_ROLE
  2016 exception INTERPRET_INFERENCE
  2017 
  2018 (*Failure reports can be adjusted to avoid interrupting
  2019   an overall reconstruction process*)
  2020 fun fail ctxt x =
  2021   if unexceptional_reconstruction ctxt then
  2022     (warning x; raise INTERPRET_INFERENCE)
  2023   else error x
  2024 
  2025 fun interpret_leo2_inference_tac ctxt prob_name node =
  2026   let
  2027     val thy = Proof_Context.theory_of ctxt
  2028 
  2029     val _ =
  2030       if Config.get ctxt tptp_trace_reconstruction then
  2031         tracing ("interpret_inference reconstructing node" ^ node ^ " of " ^ TPTP_Problem_Name.mangle_problem_name prob_name)
  2032       else ()
  2033 
  2034     val pannot = TPTP_Reconstruct.get_pannot_of_prob thy prob_name
  2035 
  2036     fun nonfull_extcnf_combined_tac feats =
  2037       extcnf_combined_tac ctxt (SOME prob_name)
  2038        [ConstsDiff,
  2039         StripQuantifiers,
  2040         InnerLoopOnce (Break_Hypotheses :: (*FIXME RemoveRedundantQuantifications :: *) feats),
  2041         AbsorbSkolemDefs]
  2042        []
  2043 
  2044     val source_inf_opt =
  2045       AList.lookup (op =) (#meta pannot)
  2046       #> the
  2047       #> #source_inf_opt
  2048 
  2049     (*FIXME integrate this with other lookup code, or in the early analysis*)
  2050     local
  2051       fun node_is_of_role role node =
  2052         AList.lookup (op =) (#meta pannot) node |> the
  2053         |> #role
  2054         |> (fn role' => role = role')
  2055 
  2056       fun roled_dependencies_names role =
  2057         let
  2058           fun values () =
  2059             case role of
  2060                 TPTP_Syntax.Role_Definition =>
  2061                   map (apsnd Binding.dest) (#defs pannot)
  2062               | TPTP_Syntax.Role_Axiom =>
  2063                   map (apsnd Binding.dest) (#axs pannot)
  2064               | _ => raise UNSUPPORTED_ROLE
  2065           in
  2066             if is_none (source_inf_opt node) then []
  2067             else
  2068               case the (source_inf_opt node) of
  2069                   TPTP_Proof.Inference (_, _, parent_inf) =>
  2070                     map TPTP_Proof.parent_name parent_inf
  2071                     |> filter (node_is_of_role role)
  2072                     |> (*FIXME currently definitions are not
  2073                          included in the proof annotations, so
  2074                          i'm using all the definitions available
  2075                          in the proof. ideally i should only
  2076                          use the ones in the proof annotation.*)
  2077                        (fn x =>
  2078                          if role = TPTP_Syntax.Role_Definition then
  2079                            let fun values () = map (apsnd Binding.dest) (#defs pannot)
  2080                            in
  2081                              map snd (values ())
  2082                            end
  2083                          else
  2084                          map (fn node => AList.lookup (op =) (values ()) node |> the) x)
  2085                 | _ => []
  2086          end
  2087 
  2088       val roled_dependencies =
  2089         roled_dependencies_names
  2090         #> map (#3 #> Global_Theory.get_thm thy)
  2091     in
  2092       val depends_on_defs = roled_dependencies TPTP_Syntax.Role_Definition
  2093       val depends_on_axs = roled_dependencies TPTP_Syntax.Role_Axiom
  2094       val depends_on_defs_names = roled_dependencies_names TPTP_Syntax.Role_Definition
  2095     end
  2096 
  2097     fun get_binds source_inf_opt =
  2098       case the source_inf_opt of
  2099           TPTP_Proof.Inference (_, _, parent_inf) =>
  2100             maps
  2101               (fn TPTP_Proof.Parent _ => []
  2102                 | TPTP_Proof.ParentWithDetails (_, parent_details) => parent_details)
  2103               parent_inf
  2104         | _ => []
  2105 
  2106     val inference_name =
  2107       case TPTP_Reconstruct.inference_at_node thy prob_name (#meta pannot) node of
  2108           NONE => fail ctxt "Cannot reconstruct rule: no information"
  2109         | SOME {inference_name, ...} => inference_name
  2110     val default_tac = HEADGOAL (blast_tac ctxt)
  2111   in
  2112     case inference_name of
  2113       "fo_atp_e" =>
  2114         HEADGOAL (Metis_Tactic.metis_tac [] ATP_Problem_Generate.combs_or_liftingN ctxt [])
  2115         (*NOTE To treat E as an oracle use the following line:
  2116         HEADGOAL (etac (oracle_based_reconstruction_tac ctxt prob_name node))
  2117         *)
  2118     | "copy" =>
  2119          HEADGOAL
  2120           (atac
  2121            ORELSE'
  2122               (rtac @{thm polarise}
  2123                THEN' atac))
  2124     | "polarity_switch" => nonfull_extcnf_combined_tac [Polarity_switch]
  2125     | "solved_all_splits" => solved_all_splits_tac
  2126     | "extcnf_not_pos" => nonfull_extcnf_combined_tac [Not_pos]
  2127     | "extcnf_forall_pos" => nonfull_extcnf_combined_tac [Universal]
  2128     | "negate_conjecture" => fail ctxt "Should not handle negate_conjecture here"
  2129     | "unfold_def" => unfold_def_tac ctxt depends_on_defs
  2130     | "extcnf_not_neg" => nonfull_extcnf_combined_tac [Not_neg]
  2131     | "extcnf_or_neg" => nonfull_extcnf_combined_tac [Or_neg]
  2132     | "extcnf_equal_pos" => nonfull_extcnf_combined_tac [Equal_pos]
  2133     | "extcnf_equal_neg" => nonfull_extcnf_combined_tac [Equal_neg]
  2134     | "extcnf_forall_special_pos" =>
  2135          nonfull_extcnf_combined_tac [Forall_special_pos]
  2136          ORELSE HEADGOAL (blast_tac ctxt)
  2137     | "extcnf_or_pos" => nonfull_extcnf_combined_tac [Or_pos]
  2138     | "extuni_bool2" => nonfull_extcnf_combined_tac [Extuni_Bool2]
  2139     | "extuni_bool1" => nonfull_extcnf_combined_tac [Extuni_Bool1]
  2140     | "extuni_dec" =>
  2141         HEADGOAL atac
  2142         ORELSE nonfull_extcnf_combined_tac [Extuni_Dec]
  2143     | "extuni_bind" => nonfull_extcnf_combined_tac [Extuni_Bind]
  2144     | "extuni_triv" => nonfull_extcnf_combined_tac [Extuni_Triv]
  2145     | "extuni_flex_rigid" => nonfull_extcnf_combined_tac [Extuni_FlexRigid]
  2146     | "prim_subst" => nonfull_extcnf_combined_tac [Assumption]
  2147     | "bind" =>
  2148         let
  2149           val ordered_binds = get_binds (source_inf_opt node)
  2150         in
  2151           bind_tac ctxt prob_name ordered_binds
  2152         end
  2153     | "standard_cnf" => HEADGOAL (standard_cnf_tac ctxt)
  2154     | "extcnf_forall_neg" =>
  2155         nonfull_extcnf_combined_tac
  2156          [Existential_Var(* , RemoveRedundantQuantifications *)] (*FIXME RemoveRedundantQuantifications*)
  2157     | "extuni_func" =>
  2158         nonfull_extcnf_combined_tac [Extuni_Func, Existential_Var]
  2159     | "replace_leibnizEQ" => nonfull_extcnf_combined_tac [Assumption]
  2160     | "replace_andrewsEQ" => nonfull_extcnf_combined_tac [Assumption]
  2161     | "split_preprocessing" =>
  2162          (REPEAT (HEADGOAL (split_simp_tac ctxt)))
  2163          THEN TRY (PRIMITIVE (Conv.fconv_rule Drule.eta_long_conversion))
  2164          THEN HEADGOAL atac
  2165 
  2166     (*FIXME some of these could eventually be handled specially*)
  2167     | "fac_restr" => default_tac
  2168     | "sim" => default_tac
  2169     | "res" => default_tac
  2170     | "rename" => default_tac
  2171     | "flexflex" => default_tac
  2172     | other => fail ctxt ("Unknown inference rule: " ^ other)
  2173   end
  2174 *}
  2175 
  2176 ML {*
  2177 fun interpret_leo2_inference ctxt prob_name node =
  2178   let
  2179     val thy = Proof_Context.theory_of ctxt
  2180     val pannot = TPTP_Reconstruct.get_pannot_of_prob thy prob_name
  2181 
  2182     val (inference_name, inference_fmla) =
  2183       case TPTP_Reconstruct.inference_at_node thy prob_name (#meta pannot) node of
  2184           NONE => fail ctxt "Cannot reconstruct rule: no information"
  2185         | SOME {inference_name, inference_fmla, ...} =>
  2186             (inference_name, inference_fmla)
  2187 
  2188     val proof_outcome =
  2189       let
  2190         fun prove () =
  2191           Goal.prove ctxt [] [] inference_fmla
  2192            (fn pdata => interpret_leo2_inference_tac
  2193             (#context pdata) prob_name node)
  2194       in
  2195         if informative_failure ctxt then SOME (prove ())
  2196         else try prove ()
  2197       end
  2198 
  2199   in case proof_outcome of
  2200       NONE => fail ctxt (Pretty.string_of
  2201         (Pretty.block
  2202           [Pretty.str ("Failed inference reconstruction for '" ^
  2203             inference_name ^ "' at node " ^ node ^ ":\n"),
  2204            Syntax.pretty_term ctxt inference_fmla]))
  2205     | SOME thm => thm
  2206   end
  2207 *}
  2208 
  2209 ML {*
  2210 (*filter a set of nodes based on which inference rule was used to
  2211   derive a node*)
  2212 fun nodes_by_inference (fms : TPTP_Reconstruct.formula_meaning list) inference_rule =
  2213   let
  2214     fun fold_fun n l =
  2215       case TPTP_Reconstruct.node_info fms #source_inf_opt n of
  2216           NONE => l
  2217         | SOME (TPTP_Proof.File _) => l
  2218         | SOME (TPTP_Proof.Inference (rule_name, _, _)) =>
  2219             if rule_name = inference_rule then n :: l
  2220             else l
  2221   in
  2222     fold fold_fun (map fst fms) []
  2223   end
  2224 *}
  2225 
  2226 
  2227 section "Importing proofs and reconstructing theorems"
  2228 
  2229 ML {*
  2230 (*Preprocessing carried out on a LEO-II proof.*)
  2231 fun leo2_on_load (pannot : TPTP_Reconstruct.proof_annotation) thy =
  2232   let
  2233     val ctxt = Proof_Context.init_global thy
  2234     val dud = ("", Binding.empty, @{term False})
  2235     val pre_skolem_defs =
  2236       nodes_by_inference (#meta pannot) "extcnf_forall_neg" @
  2237        nodes_by_inference (#meta pannot) "extuni_func"
  2238       |> map (fn x =>
  2239               (interpret_leo2_inference ctxt (#problem_name pannot) x; dud)
  2240                handle NO_SKOLEM_DEF (s, bnd, t) => (s, bnd, t))
  2241       |> filter (fn (x, _, _) => x <> "") (*In case no skolem constants were introduced in that inference*)
  2242     val skolem_defs = map (fn (x, y, _) => (x, y)) pre_skolem_defs
  2243     val thy' =
  2244       fold (fn skolem_def => fn thy =>
  2245              let
  2246                val ((s, thm), thy') = Thm.add_axiom_global skolem_def thy
  2247                (* val _ = warning ("Added skolem definition " ^ s ^ ": " ^  @{make_string thm}) *) (*FIXME use of make_string*)
  2248              in thy' end)
  2249        (map (fn (_, y, z) => (y, z)) pre_skolem_defs)
  2250        thy
  2251   in
  2252     ({problem_name = #problem_name pannot,
  2253       skolem_defs = skolem_defs,
  2254       defs = #defs pannot,
  2255       axs = #axs pannot,
  2256       meta = #meta pannot},
  2257      thy')
  2258   end
  2259 *}
  2260 
  2261 ML {*
  2262 (*Imports and reconstructs a LEO-II proof.*)
  2263 fun reconstruct_leo2 path thy =
  2264   let
  2265     val prob_file = Path.base path
  2266     val dir = Path.dir path
  2267     val thy' = TPTP_Reconstruct.import_thm true [dir, prob_file] path leo2_on_load thy
  2268     val ctxt =
  2269       Context.Theory thy'
  2270       |> Context.proof_of
  2271     val prob_name =
  2272       Path.implode prob_file
  2273       |> TPTP_Problem_Name.parse_problem_name
  2274     val theorem =
  2275       TPTP_Reconstruct.reconstruct ctxt
  2276        (TPTP_Reconstruct.naive_reconstruct_tac ctxt interpret_leo2_inference)
  2277        prob_name
  2278   in
  2279     (*NOTE we could return the theorem value alone, since
  2280        users could get the thy value from the thm value.*)
  2281     (thy', theorem)
  2282   end
  2283 *}
  2284 
  2285 end